Geophysical Data Analysis: Discrete Inverse Theory MATLAB Edition PDF

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Geophysical Data Analysis: Discrete Inverse Theory MATLAB Edition William Menke Lamont-Doherty Earth Observatory and Department of Earth and Environmental Sciences Columbia University Palisades, New York AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO ...

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Geophysical Data Analysis: Discrete Inverse Theory MATLAB Edition

William Menke

Lamont-Doherty Earth Observatory and Department of Earth and Environmental Sciences Columbia University Palisades, New York

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier 225 Wyman Street, Waltham, MA 02451, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Third edition 2012 Copyright # Elsevier Inc. 2012, 1989, 1984. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording, or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (þ 44) (0) 1865 843830; fax (þ 44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/ permissions, and selecting Obtaining permission to use Elsevier material.

Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. Library of Congress Cataloging-in-Publication Data Menke, William. Geophysical data analysis : discrete inverse theory / William Menke. – MatLab ed., 3rd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-12-397160-9 (hardback) 1. Geophysics–Measurement. 2. Oceanography–Measurement. 3. Inverse problems (Differential equations)–Numerical solutions. I. Title. QC802.A1M46 2012 551–dc23 2012000457 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our web site at store.elsevier.com Printed and bound in China 12 13 14 15 16 10 ISBN: 978-0-12-397160-9

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Dedication

To my parents William and Lee Menke who taught me how to learn

Preface

For now we see through a glass, darkly, but then . . . Paul of Tarsus

Every researcher in the applied sciences who has analyzed data has practiced inverse theory. Inverse theory is simply the set of methods used to extract useful inferences about the world from physical measurements. The fitting of a straight line to data involves a simple application of inverse theory. Tomography, popularized by the physician’s CT and MRI scanners, uses it on a more sophisticated level. The study of inverse theory, however, is more than the cataloging of methods of data analysis. It is an attempt to organize these techniques, to bring out their underlying similarities and pin down their differences, and to deal with the fundamental question of the limits of information that can be gleaned from any given data set. Physical properties fall into two general classes: those that can be described by discrete parameters (e.g., the mass of the earth or the position of the atoms in a protein molecule) and those that must be described by continuous functions (e.g., temperature over the face of the earth or electric field intensity in a capacitor). Inverse theory employs different mathematical techniques for these two classes of parameters: the theory of matrix equations for discrete parameters and the theory of integral equations for continuous functions. Being introductory in nature, this book deals mainly with “discrete inverse theory,” that is, the part of the theory concerned with parameters that either are truly discrete or can be adequately approximated as discrete. By adhering to these limitations, inverse theory can be presented on a level that is accessible to most first-year graduate students and many college seniors in the applied sciences. The only mathematics that is presumed is a working knowledge of the calculus and linear algebra and some familiarity with general concepts from probability theory and statistics. Nevertheless, the treatment in this book is in no sense simplified. Realistic examples, drawn from the scientific literature, are used to illustrate the various techniques. Since in practice the solutions to most inverse problems require substantial computational effort, attention is given to how realistic problems can be solved.

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The treatment of inverse theory in this book is divided into four parts. Chapters 1 and 2 provide a general background, explaining what inverse problems are and what constitutes their solution as well as reviewing some of the basic concepts from linear algebra and probability theory that will be applied throughout the text. Chapters 3–7 discuss the solution of the canonical inverse problem: the linear problem with Gaussian statistics. This is the best understood of all inverse problems, and it is here that the fundamental notions of uncertainty, uniqueness, and resolution can be most clearly developed. Chapters 8–11 extend the discussion to problems that are non-Gaussian, nonlinear, and continuous. Chapters 12–13 provide examples of the use of inverse theory and a discussion of the steps that must be taken to solve inverse problems on a computer. MatLab scripts are used throughout the book as a means of communicating how the formulas of inverse theory can be used in computer-based data processing scenarios. MatLab is a commercial software product of The MathWorks, Inc. and is widely used in university settings as an environment for scientific computing. All of the book’s examples, its recommended homework problems, and the case studies of Chapter 12 use MatLab extensively. Further, all the MatLab scripts used in the book are made available to readers through the book’s Web site. The book is self-contained; it can be read straight through, and profitably, even by someone with no access to MatLab. But it is meant to be used in a setting where students are actively using MatLab both as an aid to studying (that is, by reproducing the examples and case studies described in the book) and as a tool for completing the recommended homework problems. Many people helped me write this book. I am very grateful to my students at Columbia University and at Oregon State University for the helpful comments they gave me during the courses I have taught on inverse theory. Mike West, of the Alaska Volcano Observatory, did much to inspire this revision of the book, by inviting me to teach a mini-course on the subject in the fall of 2009. The use of MatLab in this book parallels the usage in Environmental Data Analysis with MatLab (Menke and Menke, 2011), a data analysis textbook that I wrote with my son Joshua Menke in 2011. The many hours we spent working together on its tutorials taught us both a tremendous amount about how to use that software in a pedagogical setting. Finally, I thank the many hundreds of scientists and mathematicians whose ideas I drew upon in writing this book.

REFERENCE Menke, W., Menke, J., 2011. Environmental Data Analysis with MatLab. Academic Press, Elsevier Inc, Oxford UK, 263pp.

COMPANION WEB SITE http://www.elsevierdirect.com/v2/companion.jsp?ISBN=9780123971609.

Introduction

I.1

FORWARD AND INVERSE THEORIES

Inverse theory is an organized set of mathematical techniques for reducing data to obtain knowledge about the physical world on the basis of inferences drawn from observations. Inverse theory, as we shall consider it in this book, is limited to observations and questions that can be represented numerically. The observations of the world will consist of a tabulation of measurements, or data. The questions we want to answer will be stated in terms of the numerical values (and statistics) of specific (but not necessarily directly measurable) properties of the world. These properties will be called model parameters for reasons that will become apparent. We shall assume that there is some specific method (usually a mathematical theory or model) for relating the model parameters to the data. The question, what causes the motion of the planets? for example, is not one to which inverse theory can be applied. Even though it is perfectly scientific and historically important, its answer is not numerical in nature. On the other hand, inverse theory can be applied to the question, assuming that Newtonian mechanics applies, determine the number and orbits of the planets on the basis of the observed orbit of Halley’s comet. The number of planets and their orbital ephemerides are numerical in nature. Another important difference between these two problems is that the first asks us to determine the reason for the orbital motions, and the second presupposes the reason and asks us only to determine certain details. Inverse theory rarely supplies the kind of insight demanded by the first question; it always requires that the physical model or theory be specified beforehand. The term inverse theory is used in contrast to forward theory, which is defined as the process of predicting the results of measurements (predicting data) on the basis of some general principle or model and a set of specific conditions relevant to the problem at hand. Inverse theory, roughly speaking, addresses the reverse problem: starting with data and a general principle, theory, or quantitative model, it determines estimates of the model parameters. In the above example, predicting the orbit of Halley’s comet from the presumably well-known orbital ephemerides of the planets is a problem for forward theory. Another comparison of forward and inverse problems is provided by the phenomenon of temperature variation as a function of depth beneath the earth’s surface. Let us assume that the temperature increases linearly with depth in the xv

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earth, that is, temperature T is related to depth z by the rule T(z) ¼ az þ b, where a and b are numerical constants that we will refer to as model parameters. If one knows that a ¼ 0.1 and b ¼ 25, then one can solve the forward problem simply by evaluating the formula for any desired depth. The inverse problem would be to determine a and b on the basis of a suite of temperature measurements made at different depths in, say, a bore hole. One may recognize that this is the problem of fitting a straight line to data, which is a substantially harder problem than the forward problem of evaluating a first-degree polynomial. This brings out a property of most inverse problems: that they are substantially harder to solve than their corresponding forward problems. Forward problem : estimates of model parameters ! quantitative model ! predictions of data Inverse problem: observations of data ! quantitative model ! estimates of model parameters Note that the role of inverse theory is to provide information about unknown numerical parameters that go into the model, not to provide the model itself. Nevertheless, inverse theory can often provide a means for assessing the correctness of a given model or of discriminating between several possible models. The model parameters one encounters in inverse theory vary from discrete numerical quantities to continuous functions of one or more variables. The intercept and slope of the straight line mentioned above are examples of discrete parameters. Temperature, which varies continuously with position, is an example of a continuous function. This book deals mainly with discrete inverse theory, in which the model parameters are represented as a set of a finite number of numerical values. This limitation does not, in practice, exclude the study of continuous functions, since they can usually be adequately approximated by a finite number of discrete parameters. Temperature, for example, might be represented by its value at a finite number of closely spaced points or by a set of splines with a finite number of coefficients. This approach does, however, limit the rigor with which continuous functions can be studied. Parameterizations of continuous functions are always both approximate and, to some degree, arbitrary properties, which cast a certain amount of imprecision into the theory. Nevertheless, discrete inverse theory is a good starting place for the study of inverse theory, in general, since it relies mainly on the theory of vectors and matrices rather than on the somewhat more complicated theory of continuous functions and operators. Furthermore, careful application of discrete inverse theory can often yield considerable insight, even when applied to problems involving continuous parameters. Although the main purpose of inverse theory is to provide estimates of model parameters, the theory has a considerably larger scope. Even in cases in which the model parameters are the only desired results, there is a plethora

Introduction

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of related information that can be extracted to help determine the “goodness” of the solution to the inverse problem. The actual values of the model parameters are indeed irrelevant in cases when we are mainly interested in using inverse theory as a tool in experimental design or in summarizing the data. Some of the questions inverse theory can help answer are the following: (a) What are the underlying similarities among inverse problems? (b) How are estimates of model parameters made? (c) How much of the error in the measurements shows up as error in the estimates of the model parameters? (d) Given a particular experimental design, can a certain set of model parameters really be determined? These questions emphasize that there are many different kinds of answers to inverse problems and many different criteria by which the goodness of those answers can be judged. Much of the subject of inverse theory is concerned with recognizing when certain criteria are more applicable than others, as well as detecting and avoiding (if possible) the various pitfalls that can arise. Inverse problems arise in many branches of the physical sciences. An incomplete list might include such entries as (a) medical and seismic tomography, (b) image enhancement, (c) curve fitting, (d) earthquake location, (e) oceanographic and meteorological data assimilation, (f) factor analysis, (g) determination of earth structure from geophysical data, (h) satellite navigation, (i) mapping of celestial radio sources with interferometry, and (j) analysis of molecular structure by X-ray diffraction. Inverse theory was developed by scientists and mathematicians having various backgrounds and goals. Thus, although the resulting versions of the theory possess strong and fundamental similarities, they have tended to look, superficially, very different. One of the goals of this book is to present the various aspects of discrete inverse theory in such a way that both the individual viewpoints and the “big picture” can be clearly understood. There are perhaps three major viewpoints from which inverse theory can be approached. The first and oldest sprang from probability theory—a natural starting place for such “noisy” quantities as observations of the real world. In this version of inverse theory, the data and model parameters are treated as random variables, and a great deal of emphasis is placed on determining the probability density functions that they follow. This viewpoint leads very naturally to the analysis of error and to tests of the significance of answers.

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The second viewpoint developed from that part of the physical sciences that retains a deterministic stance and avoids the explicit use of probability theory. This approach has tended to deal only with estimates of model parameters (and perhaps with their error bars) rather than with probability density functions per se. Yet what one means by an estimate is often nothing more than the expected value of a probability density function; the difference is only one of emphasis. The third viewpoint arose from a consideration of model parameters that are inherently continuous functions. Whereas the other two viewpoints handled this problem by approximating continuous functions with a finite number of discrete parameters, the third developed methods for handling continuous functions explicitly. Although continuous inverse theory is not the primary focus of this book, many of the concepts originally developed for it have application to discrete inverse theory, especially when it is used with discretized continuous functions.

I.2 MATLAB AS A TOOL FOR LEARNING INVERSE THEORY The practice of inverse theory requires computer-based computation. A person can learn many of the concepts of inverse theory by working through short pencil-and-paper examples and by examining precomputed figures and graphs. But he or she cannot become proficient in the practice of inverse theory that way because it requires skills that can only be obtained through the experience of working with large data sets. Three goals are paramount: to develop the judgment needed to select the best solution method among many alternatives; to build confidence that the solution can be obtained even though it requires many steps; and to strengthen the critical faculties needed to assess the quality of the results. This book devotes considerable space to case studies and homework problems that provide the practical problem-solving experience needed to gain proficiency in inverse theory. Computer-based computation requires software. Many different software environments are available for the type of scientific computation that underpins data analysis. Some are more applicable and others less applicable to inverse theory problems, but among the applicable ones, none has a decisive advantage. Nevertheless, we have chosen MatLab, a commercial software product of The MathWorks, Inc. as the book’s software environment for several reasons, some having to do with its designs and other more practical. The most persuasive design reason is that its syntax fully supports linear algebra, which is needed by almost every inverse theory method. Furthermore, it supports scripts, that is, sequences of data analysis commands that are communicated in written form and which serve to document the data analysis process. Practical considerations include the following: it is a long-lived and stable product, available since the mid-1980s; implementations are available for most commonly used types of computers; its price, especially for students, is fairly modest; and it is widely used, at least, in university settings.

Introduction

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In MatLab’s scripting language, data are presented as one or more named variables (in the same sense that c and d in the formula, c ¼ pd, are named variables). Data are manipulated by typing formula that create new variables from old ones and by running scripts, that is, sequences of formulas stored in a file. Much of inverse theory is simply the application of well-known formulas to novel data, so the great advantage of this approach is that the formulas that are typed usually have a strong similarity to those printed in a textbook. Furthermore, scripts provide both a way of documenting the sequence of a formula used to analyze a particular data set and a way to transfer the overall data analysis procedure from one data set to another. However, one disadvantage is that the parallel between the syntax of the scripting language and the syntax of standard mathematical notation is nowhere near perfect. A person needs to learn to translate one into the other.

I.3

A VERY ...